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Undergraduate practice chapters

1. Undergraduate · Chapter 261

   Two-state Markov chains · Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.8

   Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.8.

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2. Undergraduate · Chapter 262

   Orthogonal projections · Vector horizontal a=0.5; Vector vertical b=1.5

   Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=0.5; Vector vertical b=1.5.

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3. Undergraduate · Chapter 263

   Diagonal eigenmodes · First eigenvalue a=-2; Second eigenvalue b=-0.5

   Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=-2; Second eigenvalue b=-0.5.

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4. Undergraduate · Chapter 264

   Matrix conditioning · Small diagonal entry a=0.65; Final data error b=0.4

   Use the current perturbation δ=0.24, so Δb=(0,0.24). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.65; Final data error b=0.4.

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5. Undergraduate · Chapter 265

   Directional derivatives · x² coefficient a=1.5; y² coefficient b=1

   Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=1.5; y² coefficient b=1.

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6. Undergraduate · Chapter 266

   Logistic differential equations · Capacity K=6; Growth rate r=0.6

   Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=6; Growth rate r=0.6.

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7. Undergraduate · Chapter 267

   Harmonic oscillators · Angular frequency ω=1.5; Initial displacement b=1.5

   Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=1.5; Initial displacement b=1.5.

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8. Undergraduate · Chapter 268

   Euler stability · Decay rate a=1.5; Step size h=0.9

   Use n=6 updates, corresponding to time t=nh=5.4. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.5; Step size h=0.9.

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9. Undergraduate · Chapter 269

   Newton root iteration · Number whose root is sought A=3; Positive starting guess b=2.5

   Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=3; Positive starting guess b=2.5.

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10. Undergraduate · Chapter 270

    Two-state Markov chains · Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.9

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.6; Transition 2 → 1: q=0.9.

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11. Undergraduate · Chapter 271

    Orthogonal projections · Vector horizontal a=1; Vector vertical b=2

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=1; Vector vertical b=2.

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12. Undergraduate · Chapter 272

    Diagonal eigenmodes · First eigenvalue a=-1.5; Second eigenvalue b=0

    Use the completed matrix transformation at the animation endpoint. Find the image of (1,1), the determinant, and the unsigned area of the square's image. Givens: First eigenvalue a=-1.5; Second eigenvalue b=0.

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13. Undergraduate · Chapter 273

    Matrix conditioning · Small diagonal entry a=0.7; Final data error b=0.5

    Use the current perturbation δ=0.3, so Δb=(0,0.3). Find the solution perturbation and the Euclidean condition number κ₂(A). Givens: Small diagonal entry a=0.7; Final data error b=0.5.

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14. Undergraduate · Chapter 274

    Directional derivatives · x² coefficient a=2; y² coefficient b=1.5

    Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=2; y² coefficient b=1.5.

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15. Undergraduate · Chapter 275

    Logistic differential equations · Capacity K=1; Growth rate r=0.8

    Evaluate at t=3 time units. Find the amount, instantaneous growth rate, and equilibrium capacity. Givens: Capacity K=1; Growth rate r=0.8.

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16. Undergraduate · Chapter 276

    Harmonic oscillators · Angular frequency ω=2; Initial displacement b=2

    Evaluate at time t=6π/5, retaining π during calculation. Find displacement, velocity, and conserved energy (y′²+ω²y²)/2. Givens: Angular frequency ω=2; Initial displacement b=2.

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17. Undergraduate · Chapter 277

    Euler stability · Decay rate a=1.75; Step size h=1

    Use n=6 updates, corresponding to time t=nh=6. Find the Euler value, exact value at the same time, and stability classification from the amplification factor. Givens: Decay rate a=1.75; Step size h=1.

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18. Undergraduate · Chapter 278

    Newton root iteration · Number whose root is sought A=4; Positive starting guess b=3

    Perform exactly n=4 updates. Find x₄, its equation residual x₄²−A, and its absolute error relative to √A. Givens: Number whose root is sought A=4; Positive starting guess b=3.

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19. Undergraduate · Chapter 279

    Two-state Markov chains · Transition 1 → 2: p=0.7; Transition 2 → 1: q=0.1

    Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.7; Transition 2 → 1: q=0.1.

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20. Undergraduate · Chapter 280

    Orthogonal projections · Vector horizontal a=1.5; Vector vertical b=2.5

    Use θ=6π/5 radians. Find both projection coordinates, the signed component along u, and the residual length. Givens: Vector horizontal a=1.5; Vector vertical b=2.5.

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